Question

find the tangents of the acute angles in the right triangle. write each answer as a fraction in simplest form and as a decimal rounded to four places.
(right triangle with right angle at t, rt=28, ts=45, rs=53)
tan r =

tan s =

Explanation:

Step1: Recall the tangent ratio

In a right triangle, the tangent of an acute angle is the ratio of the length of the opposite side to the length of the adjacent side. For angle \( R \), the opposite side is \( ST = 45 \) and the adjacent side is \( RT = 28 \). For angle \( S \), the opposite side is \( RT = 28 \) and the adjacent side is \( ST = 45 \).

Step2: Calculate \( \tan R \)

\( \tan R=\frac{\text{opposite to }R}{\text{adjacent to }R}=\frac{45}{28} \)
\( \frac{45}{28}\approx1.6071 \)

Step3: Calculate \( \tan S \)

\( \tan S=\frac{\text{opposite to }S}{\text{adjacent to }S}=\frac{28}{45} \)
\( \frac{28}{45}\approx0.6222 \)

Answer:

\( \tan R = \frac{45}{28} \approx 1.6071 \)
\( \tan S = \frac{28}{45} \approx 0.6222 \)